The theory of axisymmetric turbulence

1950 ◽  
Vol 242 (855) ◽  
pp. 557-577 ◽  
Keyword(s):  
Differential Equations ◽  
Isotropic Turbulence ◽  
Explicit Representation ◽  
Velocity Correlation ◽  
Correlation Tensor ◽  
Gauge Invariant ◽  
Preferred Direction ◽  
Von Karman ◽  
Von Kármán

The present paper completes the theory of axisymmetric tensors and forms to the extent that is needed for the development of a theory of turbulence in which symmetry about a certain preferred direction is assumed to exist. Particular attention is given to the manner in which tensors, solenoidal in one or more indices, can be derived, uniquely, in a gauge-invariant way, as the curl of a suitably defined skew tensor. The explicit representation of the fundamental velocity correlation tensor ( ) in terms of two defining scalars is found; and the differential equations governing these scalars is also derived. In the theory of axisymmetric turbulence these latter equations replace the equation of von Karman & Howarth in the theory of isotropic turbulence.

Description of transient states of von Kármán vortex streets by low‐dimensional differential equations

1990 ◽  
Vol 2 (4) ◽  
pp. 479-481 ◽  
Author(s):  
F. Ohle ◽  
P. Lehmann ◽  
E. Roesch ◽  
H. Eckelmann ◽  
A. Hübler
Keyword(s):  
Differential Equations ◽  
Transient States ◽  
Von Karman ◽  
Vortex Streets ◽  
Karman Vortex ◽  
Low Dimensional ◽  
Von Kármán

scholarly journals The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
V. N. Grebenev ◽  
A. N. Grishkov ◽  
M. Oberlack
Keyword(s):  
Correlation Function ◽  
Asymptotic Expansion ◽  
Lie Algebra ◽  
Degrees Of Freedom ◽  
Isotropic Turbulence ◽  
Central Charge ◽  
Riemannian Metrics ◽  
Equivalence Transformation ◽  
Velocity Correlation ◽  
Correlation Tensor

The extended symmetry of the functional of length determined in an affine spaceK3of the correlation vectors for homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variablet) of the velocity fluctuations is used to equip this space by a family of the pseudo-Riemannian metricsdl2(t)(Grebenev and Oberlack (2011)). First, we observe the results obtained by Grebenev and Oberlack (2011) and Grebenev et al. (2012) about a geometry of the correlation spaceK3and expose the Lie algebra associated with the equivalence transformation of the above-mentioned functional for the quadratic formdlD22(t)generated bydl2(t)which is similar to the Lie algebra constructed by Grebenev et al. (2012). Then, using the properties of this Lie algebra, we show that there exists a nontrivial central extension wherein the central charge is defined by the same bilinear skew-symmetric formcas for the Witt algebra which measures the number of internal degrees of freedom of the system. For the applications in turbulence, as the main result, we establish the asymptotic expansion of the transversal correlation function for large correlation distances in the frame ofdlD22(t).

scholarly journals A Lagrangian direct-interaction approximation for homogeneous isotropic turbulence

1997 ◽  
Vol 345 ◽  
pp. 307-345 ◽  
Author(s):  
SHIGEO KIDA ◽  
SUSUMU GOTO
Keyword(s):  
Energy Spectrum ◽  
Differential Equations ◽  
Isotropic Turbulence ◽  
Longitudinal Velocity ◽  
Direct Interaction ◽  
Homogeneous Isotropic Turbulence ◽  
Velocity Correlation ◽  
Stationary Turbulence ◽  
Direct Interaction Approximation ◽  
Velocity Derivative

A set of integro-differential equations in the Lagrangian renormalized approximation (Kaneda 1981) is rederived by applying a perturbation method developed by Kraichnan (1959), which is based upon an extraction of direct interactions among Fourier modes of a velocity field and was applied to the Eulerian velocity correlation and response functions, to the Lagrangian ones for homogeneous isotropic turbulence. The resultant set of integro-differential equations for these functions has no adjustable free parameters. The shape of the energy spectrum function is determined numerically in the universal range for stationary turbulence, and in the whole wavenumber range in a similarly evolving form for the freely decaying case. The energy spectrum in the universal range takes the same shape in both cases, which also agrees excellently with many measurements of various kinds of real turbulence as well as numerical results obtained by Gotoh et al. (1988) for a decaying case as an initial value problem. The skewness factor of the longitudinal velocity derivative is calculated to be −0.66 for stationary turbulence. The wavenumber dependence of the eddy viscosity is also determined.

SPATIO-TEMPORAL CHAOS AND SOLITONS EXHIBITED BY VON KÁRMÁN MODEL

2002 ◽  
Vol 12 (07) ◽  
pp. 1465-1513 ◽  
Author(s):  
J. AWREJCEWICZ ◽  
V. A. KRYSKO ◽  
A. V. KRYSKO
Keyword(s):  
Differential Equations ◽  
Forced Oscillations ◽  
Phase Portraits ◽  
Algebraic Equations ◽  
Relaxation Methods ◽  
Von Karman ◽  
Flexible Plates ◽  
Spatio Temporal ◽  
Fourth Order Method ◽  
Von Kármán

Forced oscillations of flexible plates with a longitudinal, time dependent load acting on one plate side are investigated. Regular (harmonic, subharmonic and quasi-periodic) and irregular (chaotic) oscillations appear depending on the system parameters as well as initial and boundary conditions. In order to achieve highly reliable results, an effective algorithm has been applied to convert a problem of finding solutions to the hybrid type partial differential equations (the so-called von Kármán form) to that of the ordinary differential equations (ODEs) and algebraic equations (AEs). The obtained equations are solved using finite difference method with the approximations 0(h4) and 0(h2) (in respect to the spatial coordinates). The ODEs are solved using the Runge–Kutta fourth order method, whereas the AEs are solved using either the Gauss or relaxation methods. The analysis and identification of spatio-temporal oscillations are carried out by investigation of the series wij(t), wt,ij(t), phase portraits wt,ij (wij) and wtt,ij(wt,ij, wij) and the mode portraits in the planes wx,ij(wij), wy,ij (wij) and in the space wxx(wx,ij,wij), FFT as well as the Poincaré sections and pseudo-sections.

scholarly journals A note on the non-inertial similarity solution for von Kármán swirling flow

2020 ◽  
pp. 2150030
Author(s):  
Madeleine L. Combrinck
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Similarity Solution ◽  
Shooting Method ◽  
Swirling Flow ◽  
Boundary Layer Equations ◽  
Von Karman ◽  
Von Kármán

This note proposes a non-inertial similarity solution for the classic von Kármán swirling flow as perceived from the rotational frame. The solution is obtained by implementing non-inertial similarity parameters in the non-inertial boundary layer equations. This reduces the partial differential equations to a set of ordinary differential equations that is solved through an integration routine and shooting method.

scholarly journals EXPLICIT SERIES SOLUTION OF A CLOSURE MODEL FOR THE VON KÁRMÁN–HOWARTH EQUATION

2010 ◽  
Vol 52 (2) ◽  
pp. 179-202 ◽  
Author(s):  
ZENG LIU ◽  
MARTIN OBERLACK ◽  
VLADIMIR N. GREBENEV ◽  
SHI-JUN LIAO
Keyword(s):  
Isotropic Turbulence ◽  
Nonlinear Ordinary Differential Equation ◽  
Arbitrary Parameter ◽  
Closure Model ◽  
Von Karman ◽  
Homotopy Analysis ◽  
Constant Coefficients ◽  
Recursive Formulas ◽  
Von Kármán ◽  
Parameter Values

AbstractThe homotopy analysis method (HAM) is applied to a nonlinear ordinary differential equation (ODE) emerging from a closure model of the von Kármán–Howarth equation which models the decay of isotropic turbulence. In the infinite Reynolds number limit, the von Kármán–Howarth equation admits a symmetry reduction leading to the aforementioned one-parameter ODE. Though the latter equation is not fully integrable, it can be integrated once for two particular parameter values and, for one of these values, the relevant boundary conditions can also be satisfied. The key result of this paper is that for the generic case, HAM is employed such that solutions for arbitrary parameter values are derived. We obtain explicit analytical solutions by recursive formulas with constant coefficients, using some transformations of variables in order to express the solutions in polynomial form. We also prove that the Loitsyansky invariant is a conservation law for the asymptotic form of the original equation.

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